A polynomial of the second degree is generally called a quadratic polynomial. In elementary algebra, the quadratic formula is the solution of the quadratic equation. There are other ways to solve the quadratic equation instead of using the quadratic formula, such as factoring, completing the square, or graphing. Using the quadratic formula is often the most convenient way.

If f(x) is a quadratic polynomial, then f(x) = 0 is called a quadratic equation.

An equation in one unknown quantity in the form ax^{2} + bx + c = 0 is called quadratic equation.

**Sign of the Quadratic Expression**

We already acquainted with the general form of quadratic expression ax^{2} + bx + c now, here discuss the sign of the quadratic expression ax^{2} + bx + c = 0 (a ≠ 0).

When x be real then, the sign of the quadratic expression ax^{2} + bx + c is the same as a, except when the roots of the quadratic equation ax^{2} + bx + c = 0 (a ≠ 0) are real and unequal and x lies between them.

**Proof:**

We know the general form of quadratic equation ax^{2} + bx + c = 0 (a ≠ 0) ………………… (i)

Let α and β be the roots of the equation ax^{2} + bx + c = 0 (a ≠ 0). Then, we get

α + β = -b/a and αβ = c/a

Now, ax^{2} + bx + c = a(x^{2} + b/a x + c/a)

= a[x^{2} – (α + β)x + αβ]

= a[x(x – α) – β(x – α)]

or, ax^{2} + bx + c = a(x – α)(x – β) ………………… (ii)

**Case I:**

Let us assume that the roots α and β of equation ax^{2} + bx + c = 0 (a ≠ 0) are real and unequal and α > β. If x be real and β < x < α then,

x – α < 0 and x – β > 0

Therefore, (x – α)(x – β) < 0

Therefore, from ax^{2} + bx + c = a(x – α)(x – β) we get,

ax^{2} + bx + c > 0 when a < 0

and ax^{2} + bx + c < 0 when a > 0

Therefore, the quadratic expression ax^{2} + bx + c has a sign of opposite to that of a when the roots of ax^{2} + bx + c = 0 (a ≠ 0) are real and unequal and x lie between them.

**Case II:**

Let the roots of the equation ax^{2} + bx + c = 0 (a ≠ 0) be real and equal i.e., α = β.

Then, from ax^{2} + bx + c = a(x – α)(x – β) we have,

ax^{2} + bx + c = a(x – α)^{2} ……………. (iii)

Now, for real values of x we have, (x – α)^{2} > 0.

Therefore, from ax^{2} + bx + c = a(x – α)^{2} we clearly see that the quadratic expression ax^{2} + bx + c has the same sign as a.

**Case III:**

Let us assume α and β be real and unequal and α > β. If x is real and x < β then,

x – α < 0 (Since, x < β and β < α) and x – β < 0

(x – α)(x – β) > 0

Now, if x > α then x – α >0 and x – β > 0 ( Since, β < α)

(x – α)(x – β) > 0

Therefore, if x < β or x > α then from ax^{2} + bx + c = a(x – α)(x – β) we get,

ax^{2} + bx + c > 0 when a > 0

and ax^{2} + bx + c < 0 when a < 0

Therefore, the quadratic expression ax^{2} + bx + c has the same sign as a when the roots of the equation ax^{2} + bx + c = 0 (a ≠ 0) are real and unequal and x does not lie between them.

**Notes:**

(i) When the discriminant b^{2} – 4ac = 0 then the roots of the quadratic equation ax^{2} + bx + c = 0 are equal. Therefore, for all real x, the quadratic expression ax^{2} + bx + c becomes a perfect square when discriminant b^{2} -4ac = 0.

(ii) When a, b are c are rational and discriminant b^{2} – 4ac is a positive perfect square the quadratic expression ax^{2} + bx + c can be expressed as the product of two linear factors with rational coefficients.

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